When \(\alpha = 0\)  there is a big amount of articles about the existence, the uniqueness and the regularity of viscosity solutions for the equation \(F( D^2 u) = f\) , as the user's guide of Crandall Ishii and Lions [1]}, the famous paper of Ishii [2]}, and later the book of Caffarelli and Cabré, [3]}, and of Caffarelli [4]}, mainly devoted to the \({ \cal C}^1\)  and higher regularity .
